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October 1, 2003 / Vol. 28, No. 19 / OPTICS LETTERS 1733 Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack Jethro H. Greene and Allen Taflove Department of Electrical and Computer Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208 Received March 31, 2003 We report the initial three-dimensional finite-difference time-domain modeling of a vertically coupled photonic racetrack. The modeling reveals details of the full suite of space –time behavior of electromagnetic-wave phenomena involved in guiding, coupling, multimoding, dispersion, and radiation. This behavior is not easily obtainable by analytical or full-vector frequency-domain methods, measurements of terminal properties, or near-field scanning optical microscopy. © 2003 Optical Society of America OCIS codes: 230.3120, 230.5750, 230.7370. The f inite-difference time-domain (FDTD) solution method for Maxwell’s equations has been exten- sively applied to model and design microwave and photonic devices and circuits. 1 FDTD models of waveguiding photonic devices have used the effective refractive-index approximation to reduce large-scale three-dimensional (3-D) simulations to computation- ally tractable two-dimensional (2-D) problems. Rele- vant examples include reported FDTD models of micrometer-scale rings, racetracks, and disk reso- nators of various shapes. 1–4 However, the very nature of 2-D modeling prevents simulation of the important class of guided-wave photonic structures wherein the waveguide-to-resonator coupling occurs in the vertical direction rather than sideways in the plane of the resonator. Recent progress in parallel computing now permits direct full-wave FDTD modeling of 3-D photonic de- vices and circuits. In this regard, this Letter reports the formulation and results of what we believe to be the first fully 3-D FDTD model of vertical waveguide coupling and subsequent optical beam confinement in a photonic racetrack. The present work extends to the vertical coupling case previous efforts that used 2-D FDTD modeling to obtain the phenomenology of side- ways coupling. 1,2 Furthermore, this Letter points the way toward routine usage of 3-D FDTD modeling for designing large, complex, micro-optical circuits. Such modeling can reveal the full suite of space–time be- havior of electromagnetic-wave phenomena involved in guiding, coupling, multimoding, dispersion, and radia- tion, which is not easily obtainable by analytical or full-vector frequency-domain methods, 5 measurements of device terminal properties, or near-field scanning optical microscopy. 6 Referring to Figs. 1 and 2, we note that the geome- try considered here consists of a straight bus wave- guide located 1.343 mm below and parallel to one of the straight sections of a photonic racetrack that spans 64.0 mm 3 17.6 mm. Figure 1 shows the x y planes containing the bus waveguide and the racetrack, and Fig. 2 shows the y z plane cross section at P 2 , the center of one of the straight sections of the racetrack. The bus waveguide consists of a 2.015 mm 3 0.775 mm rectangular cross-section channel of refractive index n 3.33 embedded within a uniform substrate of in- dex n 3.17. The racetrack consists of a 2.015 mm 3 0.775 mm rectangular cross-section channel of refrac- tive index n 3.33 sandwiched above and below by rectangular cross-section channels of refractive index n 3.17, all embedded within a uniform superstrate of index n 1.5. By use of the bootstrapping technique, 1 the straight bus waveguide is excited in its fundamental TE mode with a numerical Gaussian pulsed source having a cen- ter free-space wavelength l 0 2.32 mm. Because the bootstrapping runs indicate 3- and 1.5-mm l 0 cutoffs for the fundamental and the first higher-order modes, respectively, the source spectrum is adjusted to span 2.17 ,l 0 , 2.47 mm (full width at half-maximum) to avoid multimoding in the straight waveguide. Wideband spectral responses are obtained with only a single FDTD run by use of concurrent discrete Fourier transformations of calculated field-versus- time waveforms 1 at the key transverse cross sections P 1 through P 4 in Fig. 1. This permits calculating as a Fig. 1. Horizontal (x y ) cuts through the 3-D FDTD mod- eling geometry: (a) through the bus waveguide beneath the racetrack, (b) through the racetrack. Vertical trans- verse field observation cut planes P 1 , P 2 , P 3 , and P 4 are also shown. 0146-9592/03/191733-03$15.00/0 © 2003 Optical Society of America

Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack

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Page 1: Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack

October 1, 2003 / Vol. 28, No. 19 / OPTICS LETTERS 1733

Initial three-dimensional finite-difference time-domainphenomenology study of the transient

response of a large vertically coupled photonic racetrack

Jethro H. Greene and Allen Taflove

Department of Electrical and Computer Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208

Received March 31, 2003

We report the initial three-dimensional finite-difference time-domain modeling of a vertically coupled photonicracetrack. The modeling reveals details of the full suite of space –time behavior of electromagnetic-wavephenomena involved in guiding, coupling, multimoding, dispersion, and radiation. This behavior is not easilyobtainable by analytical or full-vector frequency-domain methods, measurements of terminal properties, ornear-field scanning optical microscopy. © 2003 Optical Society of America

OCIS codes: 230.3120, 230.5750, 230.7370.

The finite-difference time-domain (FDTD) solutionmethod for Maxwell’s equations has been exten-sively applied to model and design microwave andphotonic devices and circuits.1 FDTD models ofwaveguiding photonic devices have used the effectiverefractive-index approximation to reduce large-scalethree-dimensional (3-D) simulations to computation-ally tractable two-dimensional (2-D) problems. Rele-vant examples include reported FDTD models ofmicrometer-scale rings, racetracks, and disk reso-nators of various shapes.1 –4 However, the verynature of 2-D modeling prevents simulation of theimportant class of guided-wave photonic structureswherein the waveguide-to-resonator coupling occursin the vertical direction rather than sideways in theplane of the resonator.

Recent progress in parallel computing now permitsdirect full-wave FDTD modeling of 3-D photonic de-vices and circuits. In this regard, this Letter reportsthe formulation and results of what we believe to bethe f irst fully 3-D FDTD model of vertical waveguidecoupling and subsequent optical beam confinement in aphotonic racetrack. The present work extends to thevertical coupling case previous efforts that used 2-DFDTD modeling to obtain the phenomenology of side-ways coupling.1,2 Furthermore, this Letter points theway toward routine usage of 3-D FDTD modeling fordesigning large, complex, micro-optical circuits. Suchmodeling can reveal the full suite of space–time be-havior of electromagnetic-wave phenomena involved inguiding, coupling, multimoding, dispersion, and radia-tion, which is not easily obtainable by analytical orfull-vector frequency-domain methods,5 measurementsof device terminal properties, or near-f ield scanningoptical microscopy.6

Referring to Figs. 1 and 2, we note that the geome-try considered here consists of a straight bus wave-guide located 1.343 mm below and parallel to one ofthe straight sections of a photonic racetrack that spans64.0 mm 3 17.6 mm. Figure 1 shows the x y planescontaining the bus waveguide and the racetrack, andFig. 2 shows the y z plane cross section at P2, thecenter of one of the straight sections of the racetrack.The bus waveguide consists of a 2.015 mm 3 0.775 mm

0146-9592/03/191733-03$15.00/0 ©

rectangular cross-section channel of refractive indexn � 3.33 embedded within a uniform substrate of in-dex n � 3.17. The racetrack consists of a 2.015 mm 30.775 mm rectangular cross-section channel of refrac-tive index n � 3.33 sandwiched above and below byrectangular cross-section channels of refractive indexn � 3.17, all embedded within a uniform superstrateof index n � 1.5.

By use of the bootstrapping technique,1 the straightbus waveguide is excited in its fundamental TE modewith a numerical Gaussian pulsed source having a cen-ter free-space wavelength l0 � 2.32 mm. Because thebootstrapping runs indicate 3- and 1.5-mm l0 cutoffsfor the fundamental and the f irst higher-order modes,respectively, the source spectrum is adjusted to span2.17 , l0 , 2.47 mm (full width at half-maximum) toavoid multimoding in the straight waveguide.

Wideband spectral responses are obtained withonly a single FDTD run by use of concurrent discreteFourier transformations of calculated field-versus-time waveforms1 at the key transverse cross sectionsP1 through P4 in Fig. 1. This permits calculating as a

Fig. 1. Horizontal (x y) cuts through the 3-D FDTD mod-eling geometry: (a) through the bus waveguide beneaththe racetrack, (b) through the racetrack. Vertical trans-verse field observation cut planes P1, P2, P3, and P4 arealso shown.

2003 Optical Society of America

Page 2: Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack

1734 OPTICS LETTERS / Vol. 28, No. 19 / October 1, 2003

Fig. 2. Vertical (y z) cross section through the 3-D FDTDmodeling geometry at P2 in the center of the couplingregion.

Fig. 3. FDTD-calculated absolute value of the y compo-nent of the electric field of the optical pulse within theracetrack: (a) 1x-directed propagation immediately afterthe initial coupling from the bus waveguide; (b) counter-clockwise propagation upon entering the first curvedsection, showing modal distortion; (c) 2x-directed propa-gation after exiting the first curved section, showingresidual modal-distortion effects.

function of frequency the magnitude and phase of thecorresponding time-harmonic transverse f ields, thelocal Poynting vectors, and the integrated opticalpower f low. In turn, this calculation yields thebroadband spectral characterization of the couplingefficiency, h (the total power f low through P4 nor-malized by the incident power f low through P1); thetransmittance, t (the normalized power f low throughP3), and the round-trip racetrack power retentionratio, r. Here, h includes the effect of the radiationloss that occurs at the beginning of the colocationof the bus and racetrack waveguides; t is obtainedwith the discrete Fourier transform temporal window

narrowed to exclude backcoupling from the racetrackto the bus waveguide; and r includes the effect oflosses that are due to radiation as well as backcouplingto the bus waveguide.

We apply the standard Yee-based FDTD method1

to a uniform 1500 3 440 3 138-cell Cartesian cubicmesh of resolution 51.7 nm (l0�45 at the centerwavelength of 2.32 mm). A 12-cell uniaxial perfectlymatched layer absorbing boundary condition1 is usedto terminate the FDTD mesh with wave ref lectionsbelow 0.01%. The simulation is run on a Cray T3Ewith 125 distributed-memory processors coupled withCray’s implementation of the message passing inter-face standard.

Figure 3 provides pseudo-color visualizations of theabsolute value of the FDTD-calculated y componentof the optical electric field within the racetrack atthree key points: (1) in the vicinity of P2 immediatelyafter coupling from the bus waveguide, showing the

Fig. 4. FDTD-calculated transverse field vectors in thevertical ( y z) cross section at P2 at the instant of peakexcitation within the bus waveguide: (a) electric field E,(b) magnetic f ield H. The bus and racetrack waveguidesare shown as shaded rectangles.

Fig. 5. FDTD-calculated transverse field vectors in thevertical (x z) cross section at P4 at the instant of peak exci-tation within the racetrack waveguide: (a) electric f ield E,(b) magnetic field H. The racetrack waveguide is shownas a shaded rectangle.

Page 3: Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack

October 1, 2003 / Vol. 28, No. 19 / OPTICS LETTERS 1735

Fig. 6. Relative FDTD-calculated Poynting vector in thedirection of propagation. The magnitudes are normalizedby the maximum magnitude in each plot, and the bounda-ries of the bus and racetrack waveguides are shown as solidrectangles. (a) x component of the Poynting vector alongthe vertical (y z) cross section at P2 at the instant of peakexcitation within the bus waveguide, (b) y component ofthe Poynting vector along the vertical (x z) cross sectionat P4 at the instant of peak excitation within the racetrackwaveguide.

Fig. 7. FDTD-calculated ratio of the coupled racetrackpower at P4 to the incident bus waveguide power as a func-tion of wavelength (left axis) and FDTD-calculated ratio ofthe transmitted bus waveguide power to the incident buswaveguide power as a function of wavelength (right axis).Subsequent backcouplings from the racetrack to the buswaveguide are excluded.

establishment of the fundamental propagating mode;(2) approaching P4 in the f irst curved section, show-ing the onset of multimoding; and (3) after exitingthe f irst curved section, showing residual modal-distortion effects and pulse dispersion. Figures 4–6provide visualizations of the corresponding transversefield vectors and longitudinal Poynting vector in thevertical cross sections at P2 and P4. These figuresshow that transient multimoding arises within thecurved section. This multimoding spatially distortsthe propagating optical pulse both by longitudinallybroadening it and by shifting its local power f lowtoward the outer curved waveguide boundary.

Figure 7 graphs the coupling efficiency, h, andtransmittance, t, versus wavelength. The midbandvalue of h can be seen to be the maximum, �6.5%,

Fig. 8. FDTD-calculated power retention within theracetrack as a function of wavelength. A single completepropagation around the racetrack from P4 back to P4 isconsidered.

dropping to �4.5% at l0 � 2.1 mm and �2.3% atl0 � 2.7 mm. Figure 8 shows the round-trip race-track power retention ratio, r, as observed at P4. Wenote that r doubles from �37% at l0 � 2.7 mm to�75% upon approaching l0 � 2 mm but then beginsto fall despite decreasing backcoupling to the buswaveguide and increasing f ield confinement withinthe racetrack’s straight sections. On the basis ofFigs. 3–6, we believe that this behavior of r is causedby increased radiation from the racetrack’s curvedsections as a result of transient multimoding thatshifts the optical power toward the outer waveguideboundary, where surface roughness can help to launchfreely propagating energy.

Work is ongoing to use 3-D FDTD modeling to de-velop corresponding phenomenology for the orthogonalz-polarized TE bus waveguide mode and to developquantitative data for resonant frequencies, Q factors,and free spectral ranges for both the y- and z-polarizedTE bus waveguide modes.

The authors thank Cray Research, Inc., for theirgenerous grant of Cray T3E supercomputer resources.We acknowledge the partial support of this workby Princeton Lightwave, Inc., and the suggestion ofthe problem geometry by Giora Griffel of PrincetonLightwave. J. H. Greene’s e-mail address is [email protected].

References

1. A. Taf love and S. C. Hagness, Computational Electro-dynamics: The Finite-Difference Time-Domain Method,2nd ed. (Artech House, Norwood, Mass., 2000).

2. S. C. Hagness, D. Raf izadeh, S. T. Ho, and A. Taf love,J. Lightwave Technol. 15, 2154 (1997).

3. A. Sakai and T. Baba, J. Lightwave Technol. 17, 1493(1999).

4. W.-H. Guo, Y.-Z. Huang, and Q.-M. Wang, IEEE Photon.Technol. Lett. 12, 813 (2000).

5. B. E. Little and S. T. Chu, IEEE Photon. Technol. Lett.12, 401 (2000).

6. G. H. Vander Rhodes, B. B. Goldberg, M. S. Unlu, S.-T.Chu, and B. E. Little, IEEE J. Sel. Top. Quantum Elec-tron. 6, 46 (2000).